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Transcript
• Channels & Keyframes

CSE169: Computer Animation

Instructor: Steve Rotenberg

UCSD, Spring 2016

• Animation

• Rigging and Animation

Animation

System

Pose

Rigging

System

Triangles

Renderer

N ...21

• Animation

When we speak of an animation, we refer to the data required to pose a rig over some range of time

This should include information to specify all necessary DOF values over the entire time range

Sometimes, this is referred to as a clip or even a move (as animation can be ambiguous)

• Pose Space

If a character has N DOFs, then a pose can be thought of as a point in N-dimensional pose space

An animation can be thought of as a point moving through pose space, or alternately as a fixed curve in pose space

One-shot animations are an open curve, while loop animations form a closed loop

Generally, we think of an individual animation as being a continuous curve, but theres no strict reason why we couldnt have discontinuities (cuts)

N ...21

t

• Channels

If the entire animation is an N-dimensional curve in pose space, we can separate that into N 1-dimensional curves, one for each DOF

We call these channels

A channel stores the value of a scalar function over some 1D domain (either finite or infinite)

A channel will refer to pre-recorded or pre-animated data for a DOF, and does not refer to the more general case of a DOF changing over time (which includes physics, procedural animation)

tii

• Channels

tmin tmax

Time

Value

• Channels

As a channel represents pre-recorded data, evaluating the channel for a particular value of t should always return the same result

We allow channels to be discontinuous in value, but not in time

Most of the time, a channel will be used to represent a DOF changing over time, but occasionally, we will use the same technology to relate some arbitrary variable to some other arbitrary variable (i.e., torque vs. RPM curve of an engine)

• Array of Channels

An animation can be stored as an array of channels

A simple means of storing a channel is as an array of regularly spaced samples in time

Using this idea, one can store an animation as a 2D array of floats (NumDOFs x NumFrames)

However, if one wanted to use some other means of storing a channel, they could still store an animation as an array of channels, where each channel is responsible for storing data however it wants

• Array of Poses

An alternative way to store an animation is

as an array of poses

This also forms a 2D array of floats

(NumFrames x NumDOFs)

Which is better, poses or channels?

• Poses vs. Channels

Which is better?

The bottom line:

Poses are faster

Channels are far more flexible and can

potentially use less memory

• Array of Poses

The array of poses method is about the

fastest possible way to playback animation

data

A pose (vector of floats) is exactly what

one needs in order to pose a rig

Data is contiguous in memory, and can all

be directly accessed from one address

• Array of Channels

As each channel is stored independently, they have the flexibility to take advantage of different storage options and maximize memory efficiency

Also, in an interactive editing situation, new channels can be independently created and manipulated

However, they need to be independently evaluated to access the current frame, which takes time and implies discontinuous memory access

• Poses vs. Channels

Array of poses is great if you just need to play back some relatively simple animation and you need maximum performance. This corresponds to many video games

Array of channels is essential if you want flexibility for an animation system or are interested in generality over raw performance

Array of channels can also be useful in more sophisticated game situations or in cases where memory is more critical than CPU performance (which is not uncommon)

• Channels

As the array of poses method is very

simple, theres not much more to say

Therefore, we will concentrate on

channels on their various storage and

manipulation techniques

• Temporal Continuity

Sometimes, we think of animations as having a particular frame rate (i.e., 30 fps)

Its often a better idea to think of them as being continuous in time and not tied to any particular rate. Some reasons include: Film / NTSC / PAL conversion

On-the-fly manipulation (stretching/shrinking in time)

Motion blur

Certain effects (and fast motions) may require one to be really aware of individual frames though

• Animation Storage

Regardless of whether one thinks of an animation as being continuous or as having discrete points, one must consider methods of storing animation data

Some of these methods may require some sort of temporal discretization, while others will not

Even when we do store a channel on frame increments, its still nice to think of it as a continuous function interpolating the time between frames

• Animation Class

class AnimationClip {

void Evaluate(float time,Pose &p);

};

class Channel {

float Evaluate(float time);

};

• Channel Storage

There are several ways to store channels. Most approaches fall into either storing them in a raw frame method, or as piecewise interpolating curves (keyframes)

A third alternative is as a user supplied expression, which is just an arbitrary math function. In practice, this is not too common, but can be handy in some situations.

One could also apply various interpolation schemes, but most channel methods are designed more around user interactivity

• Raw Data Formats

Sometimes, channels are stored simply as an array of values, regularly spaced in time at some frame rate

They can use linear or smoother interpolation to evaluate the curve between sample points

The values are generally floats, but could be compressed more if desired

The frame rate is usually similar to the final playback frame rate, but could be less if necessary

• Compressing Raw Channels

Rotational data can usually be compressed to 16 bits with reasonable fidelity

Translations can be compressed similarly if they dont go too far from the origin

One can also store a float min & max value per channel and store a fixed point value per frame that interpolates between min & max

Lowering the frame rate will also save a lot of space, but can only be done for smooth animations

One could use an automatic algorithm to compress each channel individually based on user specified tolerances

Raw channels can also be stored using some form of delta compression

• Keyframe Channels

• Keyframe Channel

A channel can be stored as a sequence of keyframes

Each keyframe has a time and a value and usually some information describing the tangents at that location

The curves of the individual spans between the keys are defined by 1-D interpolation (usually piecewise Hermite)

• Keyframe Channel

• Keyframe

Time

Value

Tangent In

Tangent Out

Keyframe (time,value)

• Keyframe Tangents

Keyframes are usually drawn so that the

incoming tangent points to the left (earlier in

time)

The arrow drawn is just for visual representation

and it should be remembered that if the two

arrows are exactly opposite, that actually means

the tangents are the same!

Also remember that we are only dealing with 1D

curves now, so the tangent really just a slope

• Why Use Keyframes?

Good user interface for adjusting curves

Gives the user control over the value of the DOF and the velocity of the DOF

Define a perfectly smooth function (if desired)

Can offer good compression (not always)

Every animation system offers some variation on keyframing

Video games may consider keyframes for compression purposes, even though they have a performance cost

• Animating with Keyframes

Keyframed channels form the foundation

for animating properties (DOFs) in many

commercial animation systems

Different systems use different variations

on the exact math but most are based on

some sort of cubic Hermite curves

• Curve Fitting

Keyframes can be generated automatically from sampled data such as motion capture

This process is called curve fitting, as it involves finding curves that fit the data reasonably well

Fitting algorithms allow the user to specify tolerances that define the acceptable quality of the fit

This allows two way conversion between keyframe and raw formats, although the data might get slightly distorted with each translation

• Keyframe Data Structure

class Keyframe {

float Time;

float Value;

float TangentIn,TangentOut;

char RuleIn,RuleOut; // Tangent rules

float A,B,C,D; // Cubic coefficients

}

Array

• Tangent Rules

Rather than store explicit numbers for tangents, it is often more convenient to store a rule and recompute the actual tangent as necessary

Usually, separate rules are stored for the incoming and outgoing tangents

Common rules for Hermite tangents include: Flat (tangent = 0)

Linear (tangent points to next/last key)

Smooth (automatically adjust tangent for smooth results)

Fixed (user can arbitrarily specify a value)

Remember that the tangent equals the rate of change of the DOF (or the velocity)

Note: I use v for tangents (velocity) instead of t which is used for time

• Flat Tangents

Flat tangents are particularly useful for making

slow in and slow out motions (acceleration

from a stop and deceleration to a stop)

v = 0

• Linear Tangents

(p0,t0)

v0out

v1in

(p1,t1)

01

0110

tt

ppvv inout

• Smooth Tangents

(p1,t1) v1out

v1in

02

0211

tt

ppvv outin

(p2,t2)

(p0,t0)

Keep in mind that this wont work on the first or last tangent (just use the linear rule)

• Step Tangent

Occasionally, one comes across the step tangent rule

This is a special case that just forces the entire span to a

constant

This requires hacking the cubic coefficients (a=b=c=0,

d=p0)

It can only be specified on the outgoing tangent and it

nullifies whatever rule is on the next incoming tangent

• Cubic Coefficients

Keyframes are stored in order of their time

Between every two successive keyframes is a span of a cubic curve

The span is defined by the value of the two keyframes and the outgoing tangent of the first and incoming tangent of the second

Those 4 values are multiplied by the Hermite basis matrix and converted to cubic coefficients for the span

For simplicity, the coefficients can be stored in the first keyframe for each span

• Cubic Equation (1 dimensional)

dctbtattf 23 cbtatdt

df 23 2

d

c

b

a

ttttf 123

d

c

b

a

ttdt

df0123 2

• Hermite Curve (1D)

v1

p1 p0

v0

t0=0 t1=1

• Hermite Curves

We want the value of the curve at t=0 to be f(0)=p0, and

at t=1, we want f(1)=p1

We want the derivative of the curve at t=0 to be v0, and

v1 at t=1

cbacbavf

ccbavf

ddcbapf

23 1213 1

0203 0

111 1

000 0

2

1

2

0

23

1

23

0

• Hermite Curves

d

c

b

a

v

v

p

p

cbav

cv

dcbap

dp

0123

0100

1111

1000

23

1

0

1

0

1

0

1

0

• Matrix Form of Hermite Curve

1

0

1

0

1

0

1

0

1

0001

0100

1233

1122

0123

0100

1111

1000

v

v

p

p

d

c

b

a

v

v

p

p

d

c

b

a

• Matrix Form of Hermite Curve

Remember, this assumes that t varies from 0 to 1

ct

gBt

tf

tf HrmHrm

1

0

1

0

23

0001

0100

1233

1122

1

v

v

p

p

ttttf

• Inverse Linear Interpolation

If t0 is the time at the first key and t1 is the time of the second key, a linear interpolation of those times by parameter u would be:

The inverse of this operation gives us:

This gives us a 01 value on the span where we now will evaluate the cubic equation

Note: 1/(t1-t0) can be precomputed for each span

1010 1,, uttuttuLerpt

01

010 ,,

tt

tttttInvLerpu

• Evaluating Cubic Spans

Tangents are generally expressed as a

slope of value/time

To normalize the spans to the 01 range,

we need to correct the tangents

So we must scale them by (t1-t0)

• Precomputing Constants

For each span we pre-compute the cubic

coefficients:

101

001

1

0

0001

0100

1233

1122

vtt

vtt

p

p

d

c

b

a

• Computing Cubic Coefficients

Note: My matrix34 class wont do this properly!

Actually, all of the 1s and 0s in the matrix make

it pretty easy to multiply it out by hand

101

001

1

0

0001

0100

1233

1122

vtt

vtt

p

p

d

c

b

a

• Evaluating the Cubic

To evaluate the cubic equation for a span,

we must first turn our time t into a 0..1

value for the span (well call this

parameter u)

aubucuddcubuaux

tt

tttttInvLerpu

23

01

010 ,,

• Channel::Precompute()

The two main setup computations a keyframe channel needs to perform are: Compute tangents from rules

Compute cubic coefficients from tangents & other data

This can be done in two separate passes through the keys or combined into one pass (but keep in mind there is some slightly tricky dependencies on the order that data must be processed if done in one pass)

• Extrapolation Modes

Channels can specify extrapolation modes to define how the curve is extrapolated before tmin and after tmax

Usually, separate extrapolation modes can be set for before and after the actual data

Common choices: Constant value (hold first/last key value)

Linear (use tangent at first/last key)

Cyclic (repeat the entire channel)

Cyclic Offset (repeat with value offset)

Bounce (repeat alternating backwards & forwards)

• Extrapolation

Note that extrapolation applies to the

entire channel and not to individual keys

In fact, extrapolation is not directly tied to

keyframing and can be used for any

method of channel storage (raw)

• Extrapolation

Flat:

Linear:

tmin

tmax

• Extrapolation

Cyclic:

Cyclic Offset:

• Extrapolation

Bounce:

• Keyframe Evaluation

The main runtime function for a channel is something like:

float Channel::Evaluate(float time);

This function will be called many times

For an input time t, there are 4 cases to consider: t is before the first key (use extrapolation)

t is after the last key (use extrapolation)

t falls exactly on some key (return key value)

t falls between two keys (evaluate cubic equation)

• Channel::Evaluate()

The Channel::Evaluate function needs to

be very efficient, as it is called many times

while playing back animations

There are two main components to the

evaluation:

Find the proper span

Evaluate the cubic equation for the span

• Random Access

To evaluate a channel at some arbitrary time t, we need to first find the proper span of the channel and then evaluate its equation

As the keyframes are irregularly spaced, this means we have to search for the right one

If the keyframes are stored as a linked list, there is little we can do except walk through the list looking for the right span

If they are stored in an array, we can use a binary search, which should do reasonably well

• Finding the Span: Binary Search

A very reasonable way to find the key is by a binary search. This allows pretty fast (log N) access time with no additional storage cost (assuming keys are stored in an array (rather than a list))

Binary search is sometimes called divide and conquer or bisection

For even faster access, one could use hashing algorithms, but that is probably not necessary, as they require additional storage and most real channel accesses can take advantage of coherence (sequential access)

• Finding the Span: Linear Search

One can always just loop through the keys from the beginning and look for the proper span

This is an acceptable place to start, as it is important to get things working properly before focusing on optimization

It may also be a reasonable option for interactive editing tools that would require key frames to be stored in a linked list

Of course, a bisection algorithm can probably be written in less than a dozen lines of code

• Sequential Access

If a character is playing back an animation, then it will be accessing the channel data sequentially

Doing a binary search for each channel evaluation for each frame is not efficient for this

If we keep track of the most recently accessed key for each channel, then it is extremely likely that the next access will require either the same key or the very next one

This makes sequential access of keyframes potentially very fast

However there is a catch

• Sequential Access

Consider a case where we have a video game with 20 bad guys running around

They all need to access the same animation data (which should only be stored once obviously)

However, they might each be accessing the channels with a different time

Therefore, the higher level code that plays animations needs to keep track of the most recent keys rather than the simpler solution of just having each channel just store a pointer to its most recent key

Thus, the animation player class needs to do considerable bookkeeping, as it will need to track a most recent key for every channel in the animation

• High Performance Channels

If coefficients are stored, we can evaluate the cubic equation with 4 additions and 4 multiplies

In fact, a,b,c, & d can actually be precomputed to include the correction for 1/(t1-t0) so that the cubic can be directly solved for the original t. This reduces it to 3+ and 3*

In other words, evaluating the cubic is practically instantaneous, while jumping around through memory trying to locate the span is far worse

If we can take advantage of sequential access (which we usually can), we can reduce the span location to a very small number of operations

• Robustness

The channel should always return some reasonable value regardless of what time t was passed in If there are no keys in the channel, it should just return 0

If there is just 1 key, it should return the value of that key

If there are more than 1 key, it should evaluate the curve or use an extrapolation rule if t is outside of the range

At a minimum, the constant extrapolation rule should be used, which just returns the value of the first (or last) key if t is before (or after) the keyframe range

When creating new keys or modifying the time of a key, one needs to verify that its time stays between the key before and after it

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